$f_i(\vec{x},t)=w_if(\vec{x},\vec{v}_i,t)$

f_i(\vec{x},t)=w_if(\vec{x},\vec{v}_i,t)

$f^{(0)}(\vec{x},\vec{v},t)=c(\vec{x},t)\left(\displaystyle\frac{m\beta}{2\pi}\right)^{3/2}e^{-\beta m(\vec{v}-\vec{u}(\vec{x},t))^2/2}$

f^{(0)}(\vec{x},\vec{v},t)=c(\vec{x},t)\left(\displaystyle\frac{m\beta}{2\pi}\right)^{3/2}e^{-\beta m(\vec{v}-\vec{u}(\vec{x},t))^2/2}

$\displaystyle\frac{df}{dt}=-\displaystyle\frac{1}{\tau}(f-f^{(0)})$

\displaystyle\frac{df}{dt}=-\displaystyle\frac{1}{\tau}(f-f^{(0)})

$f_i^{eq}=\rho\omega_i\left(1+\displaystyle\frac{3\vec{u}\cdot\vec{e}_i}{c}+\displaystyle\frac{9(\vec{u}\cdot\vec{e}_i)^2}{2c^2}-\displaystyle\frac{3u^2}{2c^2}\right)$

f_i^{eq}=\rho\omega_i\left(1+\displaystyle\frac{3\vec{u}\cdot\vec{e}_i}{c}+\displaystyle\frac{9(\vec{u}\cdot\vec{e}_i)^2}{2c^2}-\displaystyle\frac{3u^2}{2c^2}\right)

$f_i(\vec{x},t)\leftarrow f_i(\vec{x}+ce_i\delta t,t+\delta t)$

f_i(\vec{x},t)\leftarrow f_i(\vec{x}+ce_i\delta t,t+\delta t)